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If e were rational, then e = n/m for some positive integers m, n. So then 1/e = m/n. But the series expansion for 1/e is
1/e = 1 - 1/1! + 1/2! - 1/3! + ...
Call the first n values of this alternating series S(n). How good is this approximation to e? Well, the error is bounded by the next part of the alternating series:
0 < | 1/e - S(n) | = | m/n - S(n)| < 1/(n+1)!
But multiplying through by n!, you can see that
0 < | integer - integer | < 1/(n+1) < 1.
But there is no integer between 0 and 1, so this is a contradiction; e must be irrational.
Find the derivatives of the following functions a) y = 5x 4 +3x -1-x 3 b) y = (x+1) -1/2 c) y= e x2+1 d) y= e 3x lnx e) y =ln(x+1/x)y
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Fourier series - Partial Differential Equations One more application of series arises in the study of Partial Differential Equations. One of the more generally employed method
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Jay bought twenty-five $0.37 stamps. How much did he spend? To ?nd how much Jay spent, you must multiply the cost of each stamp ($0.37) through the number of stamps purchased (
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1. Four different written driving tests are administered by a city. One of these tests is selected at random for each applicant for a drivers license. If a group of 2 women and 4 m
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